We investigate the asymptotic behavior of weak solutions to the semilinear non-autonomous wave equation u tt -Δu+u t |u t | p-1 =V(t)u|u| p-1 +f(·,t), where V(t) is a positive time dependent potential satisfying V(t)=O((1+t) -λ ) as t→+∞ and f t decays to 0 as t→+∞. We show that for 0≤λ≤p there are initial values such that the energy norm of the corresponding solutions grows at least polynomially as t→+∞, while if λ>p the energy norm remains uniformly bounded for any choice of initial values; moreover, in certain cases there is an absorbing ball for the orbits.